Problem: The sum of two angles is $86^\circ$. Angle 2 is $106^\circ$ smaller than $3$ times angle 1. What are the measures of the two angles in degrees?
Solution: Let $x$ equal the measure of angle 1 and $y$ equal the measure of angle 2. The system of equations is then: ${x+y = 86}$ ${y = 3x-106}$ Since we already have solved for $y$ in terms of $x$ , we can use substitution to solve for $x$ and $y$ Substitute ${3x-106}$ for $y$ in the first equation. ${x + }{(3x-106)}{= 86}$ Simplify and solve for $x$ $ x+3x - 106 = 86 $ $ 4x-106 = 86 $ $ 4x = 192 $ $ x = \dfrac{192}{4} $ ${x = 48}$ Now that you know ${x = 48}$ , plug it back into $ {y = 3x-106}$ to find $y$ ${y = 3}{(48)}{ - 106}$ $y = 144 - 106$ ${y = 38}$ You can also plug ${x = 48}$ into $ {x+y = 86}$ and get the same answer for $y$ ${(48)}{ + y = 86}$ ${y = 38}$ The measure of angle 1 is $48^\circ$ and the measure of angle 2 is $38^\circ$.